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Mandelbrots
Iterating functions
Mandelbrots, named after Benoit Mandelbrot are defined by a process of iterating functions.



Above is an image of the Mandelbrot set in a complex parameter plane. The displayed portion includes the rectangle -0.500000 < x < 0.500000, -0.500000 < y < 1.500000 displayed as an image 500 pixels wide by 30 pixels high.

The Mandelbrot formula
Given a complex function, f(z), and a complex number, z(0), consider the sequence z(0), f(z(0)), f(f(z(0))), . . . known as the orbit of z(0) under f(z). The magnitude of a complex number a + bi is the real number that is the distance from the point representing the complex number and the origin. This is the square root of (a*a + b*b).

f(n)(z(0)) is defined to be the nth term in the orbit of z(0) under f(z), where f(0)(z(0)) = z(0). If there exists M > 0 such that the absolute value of f(n)(z(0)) is <= M for all n, then the orbit of z(0) under f(z) is said to be bounded. Otherwise, the orbit of z(0) under f(z) is said to escape to infinity.

Creating your own Mandelbrots
The Mandelbrots may be created by coloring the pixels at each point on the function based on the number of iterations needed for the orbit to escape to infinity.

It is possible to generate an infinite variety of Mandelbrots either by programing your computer to run the Mandelbrot iteration formula or by using one of these programs:
  1. the David Joyce Mandelbrot Applet
  2. the Eclectasy Fractal-Explorer
To use the progams, input your real and imaginary coordinates and selecting a number of iterations and colors.
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